Mathematics for the interested outsider

Examples of Specht Modules

on which acts trivially. And so is a one-dimensional space with the trivial group action. This is the only possibility anyway, since , and we’ve seen that is itself a one-dimensional vector space with the trivial action of .

Next, consider — with parts each of size . This time we again have one polytabloid. We fix the Young tableau

Since every entry is in the same column, the column-stabilizer is all of . And so we calculate the polytabloid

We conclude that is a one-dimensional space with the signum representation of . Unlike our previous example, there is a huge difference between and ; we’ve seen that is actually the left regular representation, which has dimension .

Finally, if , then we can take a tableau and write a tabloid

where the notation we’re using on the right is well-defined since each tabloid is uniquely identified by the single entry in the second row. Now, the polytabloid in this case is , since the only column rearrangement is to swap and . It’s straightforward to see that these polytabloids span the subspace of where the coefficients add up to zero:

As a basis, we can pick . We recognize this pattern from when we calculated the invariant subspaces of the defining representation of . And indeed, is the defining representation of , which contains as the analogous submodule to what we called before.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.